Constrained Planarity and Augmentation Problems

A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E). Each vertex m in T corresponds to a subset of the vertices of the graph called ``cluster''. c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c-planarity testing is unknown. It has been shown by Dahlhaus, Eades, Feng, Cohen that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In the first part of the thesis, we provide a polynomial time algorithms for c-planarity testing of specific planar clustered graphs: Graphs for which - all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each non-connected cluster its super-cluster and all its siblings in T are connected, - for all clusters m G-G(m) is connected. The algorithms are based on the concepts for the subgraph induced planar connectivity augmentation problem, also presented in this thesis. Furthermore, we give some characterizations of c-planar clustered graphs using minors and dual graphs and introduce a c-planar augmentation method. Parts II deals with edge deletion and bimodal crossing minimization. We prove that the maximum planar subgraph problem remains NP-complete even for non-planar graphs without a minor isomorphic to either K(5) or K(3,3), respectively. Further, we investigate the problem of finding a minimum weighted set of edges whose removal results in a graph without minors that are contractible onto a prespecified set of vertices. Finally, we investigate the problem of drawing a directed graph in two dimensions with a minimal number of crossings such that for every node the incoming and outgoing edges are separated consecutively in the cyclic adjacency lists. It turns out that the planarization method can be adapted such that the number of crossings can be expected to grow only slightly for practical instances

Packing Topological Minors Half-Integrally

The packing problem and the covering problem are two of the most general questions in graph theory. The Erd\H{o}s-P\'{o}sa property characterizes the cases when the optimal solutions of these two problems are bounded by functions of each other. Robertson and Seymour proved that when packing and covering $H$-minors for any fixed graph $H$, the planarity of $H$ is equivalent with the Erd\H{o}s-P\'{o}sa property. Thomas conjectured that the planarity is no longer required if the solution of the packing problem is allowed to be half-integral. In this paper, we prove that this half-integral version of Erd\H{o}s-P\'{o}sa property holds with respect to the topological minor containment, which easily implies Thomas' conjecture. Indeed, we prove an even stronger statement in which those subdivisions are rooted at any choice of prescribed subsets of vertices. Precisely, we prove that for every graph $H$, there exists a function $f$ such that for every graph $G$, every sequence $(R_v: v \in V(H))$ of subsets of $V(G)$ and every integer $k$, either there exist $k$ subgraphs $G_1,G_2,...,G_k$ of $G$ such that every vertex of $G$ belongs to at most two of $G_1,...,G_k$ and each $G_i$ is isomorphic to a subdivision of $H$ whose branch vertex corresponding to $v$ belongs to $R_v$ for each $v \in V(H)$, or there exists a set $Z \subseteq V(G)$ with size at most $f(k)$ intersecting all subgraphs of $G$ isomorphic to a subdivision of $H$ whose branch vertex corresponding to $v$ belongs to $R_v$ for each $v \in V(H)$. Applications of this theorem include generalizations of algorithmic meta-theorems and structure theorems for $H$-topological minor free (or $H$-minor free) graphs to graphs that do not half-integrally pack many $H$-topological minors (or $H$-minors)

Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth

We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bounded treewidth.Comment: Graph Drawing 201

Are there any good digraph width measures?

Several different measures for digraph width have appeared in the last few years. However, none of them shares all the "nice" properties of treewidth: First, being \emph{algorithmically useful} i.e. admitting polynomial-time algorithms for all \MS1-definable problems on digraphs of bounded width. And, second, having nice \emph{structural properties} i.e. being monotone under taking subdigraphs and some form of arc contractions. As for the former, (undirected) \MS1 seems to be the least common denominator of all reasonably expressive logical languages on digraphs that can speak about the edge/arc relation on the vertex set.The latter property is a necessary condition for a width measure to be characterizable by some version of the cops-and-robber game characterizing the ordinary treewidth. Our main result is that \emph{any reasonable} algorithmically useful and structurally nice digraph measure cannot be substantially different from the treewidth of the underlying undirected graph. Moreover, we introduce \emph{directed topological minors} and argue that they are the weakest useful notion of minors for digraphs